Electromagnetic horn



43-1186 Exam May 25, 1948. w. BARROW R 3, 03

ELECTROMAGNETIC HORN orig. Original Filed Jan. 3, 1959 10 Sheets-Sheet l INVENTOR M4 M47? 4 fimrnaw via I ATTORNEY 31 am m May 25, 1948. w BARROW Re. 23,003

ELECTROMAGNETIC HORN Original Filed Jan. 5, 1939 10 Sheets-Sheet 2 mvsmon W/uvn? 1.. BARROW ATTORNEY 1943- w. BARROW Re. 23,003

ELECTROMAGNETIC HORN Original Filed Jan. 3, 1939 10 Sheets-Sheet s I i l 4. VERTICAL APE/PTl/RE, IN X .1 ll .1 I -I I l'llll INVENTOR W/LMER L. HAM-raw av M 6 ATTORNEY txami May 25, 1948. w. L. BARROW ELECTROMAGNETIC HORN Original Filed Jan. 3, 1939 10 Sheets-Sheet 5 a .3 3 .2 S m a Q N oI .Qw 0W/ 9 oj/l// Q w .Q w L! v i M 7 lr 2 a INVENTOR W/LMER L fiAR/ww BY I z M ATTORNEY May 25, 1948. w. L. BARROW ELECTROMAGNETIC HORN Original Filed Jan. 3, 1939 10 Sheets-Sheet 6 INVENTOR VV/LMEI? L BARROW awe/w All/38 BY e.

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May 25, 1948.

W. L. BARROW ELECTROMAGNETIC HORN Original Filed Jan. 3, 1939 FAA/7E Amaze (p N 10 Sheets-Sheet 1O W/LMER A BAEEOW Reicsued May 25, 1948 ELECTROMAGNETIC HORN Wilmer L. Barrow, Concord, Mass., assignor to h Corporation, New York, N. Y., a corporation of New York Original No. 2,255,042, dated September 9, 1941,

Serial No. 249,005, January 3, 1939. Application for reissue March 21, 1942, Serial No.

66 Claims. 1

The present invention relates to electromagnetic horns.

An object of the present invention is to improve upon electromagnetic horns, particularly of sectoral or pyramidal shape, with the aid of which it shall be possible to obtain a beam of specified angular spread.

Another object is to provide a horn of the above-described character the beam of which shall be sharp.

A further object is to provide an optimum sharpness of beam.

Another object is to concentrate the radiant energy of the beam in one direction.

Still a further object is to provide an electromagnetic horn of the above-described character the beam diagram or radiation pattern of which shall have a smooth contour.

A further object is to provide the said smooth contour with a single peak representing a single maximum electric intensity.

Another object is to provide an electromagnetic horn the beam of which shall have a p ified power gain.

Still another object is to render the said gain a maximum.

Another object is to provide a horn having a predetermined radiation performance.

A further object is to provide quantitative curves to facilitate the designs or such horns.

Other and further objects will be described hereinafter and will be particularly pointed out in the appended claims.

The invention will now be described in connection with the accompanying drawings, in which Fig. 1 is a perspective, partly broken away, of a sectoral electromagnetic horn embodying the invention and fed by a hollow-pipe line;

Figs. 2 and 3 are similar perspectives of modiflcations;

Fig. 4 is a diagrammatic perspective of a sectoral horn. disposed in Cartesian and cylindrical systems of coordinates, and carrying symbols useful in describing the invention:

Fig. 5 is a section showing the field configuration or the HM wave. the section being in the plane passing longitudinally through the horn of Fig. l, and containing the principal axis of the horn and the lines 8-8, which plane may, for convenience. be referred to as the longitudinal horizontal plane. p rallel to the XZ plane of Fig. 4, and which section may, for convenience. be referred to as the longitudinal horizontal section:

Fig. 6 is a section showing the field configuration of the Hm wave, the section being in the plane at right angles to the first-named plane and passing longitudinally through the horn of Fig. 1, and containing the principal axis and the lines l-I. which plane may, for convenience. be referred to as the longitudinal vertical plane, or

2 the XY plane of Fig. 4, and which section may. for convenience, be referred to as the longitudinal vertical section;

Fig. 'l is a section showing the field configuration of the Hm wave, the section being along the vertically disposed cylindrical surface represented by the circular arc l-I of Fig. 5, which cylindrical surface 'l-l may, for convenience, be referred to a transverse cylindrical surface, parallel to the Y axis of Fig. 4, and which section may, for convenience, be referred to as a transverse vertical cylindrical section;

Fig. 3 is the same longitudinal horizontal section as Fig. 5, showing the field configuration of the Hm wave;

Fig. 9 is the same transverse vertical cylindrical section as Fig. 7, represented by the circular arc 8-9 of Fig. 8, showing the field configuration of the Hm wave;

Fig. 10 is the same longitudinal vertical section as Fig. 6. the field configuration of the Hm wave being shown;

Fig. 11 is a plot of two curves showing the relation between beam angle and the vertical dimension of the aperture of the horn, indicated in Fig. 4 by the length a in terms of the wavelength ll;

Fig. 12 is a diagram illustrating the sphericalcoordinate system and design parameters used in calculating horizontal radiation patterns;

Fig. 13 is a simplified horizontal radiation pattern for the Hm wave, showing also its beam angle;

Figs. 14 to 21 are similar horizontal radiation half-patterns for the Hm wave, corresponding to horns all having a flare angle o=30. and with their radial horn lengths pi equal to 6, 8, 10, l2, 15. 20, 30 and 50 wave-lengths, respectively, the remainder of each of these half-patterns (not shown) being mirror images thereof;

Figs. 22 to 29 are similar horizontal radiation half-patterns for the Hm wave corresponding to horns all havi g a radial length p1 equal to 12 wave-lengths. and with their flare angles o equal to 30, and respectively:

Fig. 30 is a plot of a series of curves for the Hm wave. showing the relation between beam angle and flare angle 4: for values of A equal to 6, 8, 10, 12, 15, 20. 30 and 50. respectively;

Fig. 31 is a similar plot for the Hon wave. showing the relation between beam angle and for values of the flare angle o equal to 10, 20, 30, 40 and 50', respectively;

Fig. 32 is a plot similar to Fig. 31 for the Hm wave, with the horizontal dimension of the mouth or aperture of the horn, indicated in Fig. 4 by the length b, in terms of the wave-length A, substituted for the abscissa El. A

plotted against beam angle;

Fig. 34 is a plot of a series or curves for the H wave, showing the relation between power gain, in terms of and flare angle M, for radial lengths equal to 6, 8, 10, 12, 15, 20, 30 and 50 wave-lengths, respectively;

Fig. 35 is a similar plot for the Hm wave, but with the parameter substituted for the abscissa 4o;

Fig. 36 is a plot two curves tor the Him and Him waves. showing the relation between optimum values oi the flare angle on and radial horn length for providing maximum power gain;

Figs. 37 to 44 correspond to Figs. 14 to 21, respectively. for the H1 wave, but with a constant flare angle o=35;

Figs. 45 to 52 similarly correspond to Figs. 22 to 29, respectively, for the Hip wave, but with a constant radial length and with varying flare angle die for intervals oi between 20 and 55;

Figs. 53 and 54 correspond to Figs. 34 and 35 for the Hm wave;

Fig. 55 is a perspective of a further modified horn;

Fig. 56 represents the curve of Fig. 33, for the H wave, but plotted directly, in logarithmic coordinates. in terms oi. optimum flare angle :50 and associated radial horn length -i instead of indirectly against the beam angle;

Fig. 5'7 similarly represents the two curves oi Fig. 36, for the Hm and Hm waves, respectively, but plotted directly, in logarithmic coordinates, in terms of optimum flare angle on and associated radial horn length instead of indirectly against the maximum power gain; and

Fig. 58 is a similar logarithmic plot 0! two similar curves for the respective no.1 and Hm waves. corresponding to Edge. 35 and 54, respectively, but

plotted in terms of the horizontal dimension of the mouth or aperture of the horn,

against values of the radial horn length for producing maximum power gain.

In Figs. 34, 35, 36, 53 and 54, the numerical scale of the ordinate is to be multiplied by the value of for the horn in question to get the numerical value oi the power gain.

In Figs. 1 to 4 and 55, there is illustrated a horn is or rectangular cross section, flaring smoothly and continuously from its throat or small or refleeting end, or the back of the horn, to a mouth or aperture or large end at its front. The principal or central axis of the horn extends between the smaller and larger ends of the horn. At its throat. it is shown connected to a hollow-pipe system comprising an elongated hollow pipe or tube body portion or section l8 that may extend over any desired distance from the horn, to the left, as viewed in Figs. 1 to 4 and 55.

The horn may be constituted of a formed sheet or conducting material, like metal, such as copper or aluminum. or it may be constituted of other material it its inner wall is otherwise rendered a conductor 0! the said waves. The space inside the horn, being open to the atmosphere, is naturally a non-conductor. Galvanized-iron sheeting, and thin electrolytic copper foil 2| cemented to insulating plywood l5, as illustrated in Figs. 5 to 10, have both proven satisfactory. The hollowpipe body portion H, to which the horn I6 is connected. may be of any desired material. If the horn I5 is of simple rectangular shape. the pipe portion l8, from which it flares out, may be likewise rectangular. Screen or other semi-open construction may also be employed.

Two or the sides may be substantially parallel, and the other two sides flared, as illustrated in Figs. 1. 3 and 55, or the horn may be pyramidal, with the sides of the horn flaring in tour oppositely disposed directions, as illustrated in Fig. 2. The horns of Figs. 1, 3 and 55 may be described as of sectoral shape, and the horn of Fig. 2 may be termed pyramidal. Horns provided with straight sides in longitudinal cross-section, as illustrated in Figs. 1 and 2, are economical and easy to construct. Thorough analysis of this structure has been made and applies directly to seetoral horns and indirectly to horns of pyramidal shape.

In Figs. 3 and 55, the cross-section in the X2 plane has a hyperbolic contour, concave in one case and convex in the other. In both instances, the sides of the horn approach asymptotically to straight lines which cross to form an angle between them, which is the same as the flare angle 4 of the horn of Fig. 1, because the waves in side these horns at a distance several wavelengths irom the throat are governed by the dimensions and flare angle 50 of the extended portions of the sides. The sectoral horns of Figs. 3 and 55 behave, as regards their radiation properties, Just as does the sectoral born 0! Fig. 1. The curves oi these horns, near the throat. need not be hyperbolic; substantially iami any curve, within wide limits, could be employed with substantially the same results.

The horn of the present invention may be used for transmission or reception of ultra-highfrequency electromagnetic waves. In transmission, electromagnetic energy, transmitted through the interior of the pipe or tube I8 from a projecting metal exciting or absorbing antenna rod or other translating apparatus II, is delivered to the throat of the horn and propagated through the interior of the horn to the mouth as horn waves." At the mouth, substantially all of this energy is radiated into free space as ordinary radio waves. The horn thus constitutes a directive electromagnetic radiator.

In receiving, a similar, but reverse, process takes place, the electromagnetic waves being received by the horn l6, and communicated to a receiving system (not shown).

The rod I4 is shown in Figs. 1 and 55 disposed approximately centrally in the hollow pipe I8, substantially at right angles to the axis of the horn, but it may be disposed unsymmetrically in the horn, to give a modified directive pattern for the radiant energy.

Sending apparatus (not shown) may be connected to a coaxial-line system l0, ill, or to a parallel-wire system or to any other desired connecting system. The conductor i2 may be extended into the bell of the horn or the tube l8, either parallel to the top and bottom walls as in Fig. 3, or at right angles thereto, as in Figs. 1, 2 and 55, to constitute the antenna l4.

Diflerent types of horn waves, and combinations of the same, may be separately excited and propogated within the horn, or absorbed by the horn, by properly arranging the exciting rod or rods in or out of the throat of the horn, both as to the position of, and the current in, the rod or rods. This is described more at length in my copending application, Serial No. 249,910, filed January 9, 1939.

One of the most important modes or wave types is the lowest-order transversely polarized horn wave Ho,1, with the electric vector mainly parallel everywhere to the vertical direction, obtained with an exciting rod ll transverse to the axis and in the vertical plane. Another important mode or wave type is the Hm wave. These two wave types are probably the best for sending a single beam of radiant energy and are the waves most naturally adopted for receiving. For beam transmission, a horn of rectangular cross section perpendicular to its principal or central axis and the orientation of the exciting rod perpendicular to this horn axis, oifer certain features, among them the important feature of a radiated linearly polarized space wave. The rectangular shape, furthermore, permits independent control of the width of the radiated beam in the horizontal and vertical planes.

Beams transmitted from directive antenna systems are usually accompanied by small amounts or radiation in directions other than those intended. These small amounts of radiation may be referred to as secondary lobes. Beams radiated from sectoral horns are remarkably free from secondary lobes. For this reason, the use of sectoral horns is particularly advantageous in certain kinds of applications where the shape of the beam plays an essential role in the operation. In the blind-landing of airplanes. for example, it is desirable, not only that the beam be very sharp, but also that it be peculiarly free from secondary lobes. In one such application,

a smooth straight-line intersection is formed between two overlapping beams; systems of this type are commonly referred to as equal-signal" systems. Horn radiators can provide such smooth overlapping beams without waviness or spurious components that would aifect the straightness of this path of intersection. No other types of antennas have been found to produce so smooth beams with such small secondary lobes. Patterns of this character are useful also in other applications. such as direction-finding and obstacledetection.

In Figs. 1, 3 and 55, as before described, the horn I5 is shown excited by means of a hollowpipe transmission line l8, connected to the throat of the born, with the translating apparatus positioned in the pipe I8, at the rear of the throat of the horn. The antenna. then first excites waves in the hollow pipe, which are transmitted through the pipe l8 to the horn and thence into free space as ordinary radio waves. It is sometimes desirable, however, as explained in my application, Serial No. 240,545, filed November 15, 1938, that the translating apparatus be positioned directly in the throat of the horn, as illustrated in Fig. 2, in order directly to excite the horn itself, without the use of a hollow-pipe transmission line. It then operates efllciently to receive substantially all of the incident energy, when used as a receiver; or, when used as a transmitter, to excite waves of the horn type, and thereby to radiate a beam of character appropriate to the horn rather than to the apparatus and the antenna itself. With the apparatus in the throat. furthermore, the horn has smaller physical dimensions than when connected to the hollow pipe.

The hollow-pipe method of Figs. 1, 3 and 55 is mainly useful for wave-lengths less than about 20 centimeters, and the method of Fig. 2, involving the positioning of the antenna or other energy-translation means directly in the throat of the horn, is applicable to longer waves. Substantially the same radiation pattern may be produced with either arrangement.

The invention is not, of course, limited to the use of an exciting or absorbing rod ll. Other radiating or absorbing means, such as a vacuum tube, may also be employed, as described, for example, in my said application, Serial No. 240,545, filed November 15, 1938. As is also explained in the said application, Serial No. 240,545, optimum conditions may be obtained by adjusting a piston (not shown) at the back or throat of the horn, thus to resonate or tune the throat of the horn, thereby rendering the throat of the horn more responsive to a particular frequency or a narrow band of frequencies than to other frequencies, and also in other ways.

Symbols that will be used in this specification will be understood by reference to Fig. 4. The lower of the two parallel sides of the horn of Fig. 1 is here assumed to lie in the XZ plane, with its non-parallel or flaring sides, extended, intersecting on, and making equal angles with. the Y axis. The system may be regarded as symbolizing also cylindrical co-ordinates y, p and as will be clear also from Fig. 12. The upper parallel side of the horn is parallel to the X2 plane, and at a distance a therefrom. It is assumed that the throat of the horn is disposed along a vertical cylindrical surface having a radius p0 and with its axis coincident with the Y axis. The non-parallel sides, each 01 length i-po, are symmetrically disposed in planes per- 7 pendicular to the X2 plane, forming with the XY plane a dihedral angle equal to and parallel to the lines of electric intensity of the waves propagated within the horn. The horn is regarded as having a forward direction in the positive direction of the X axis. The horizontal length of the mouth or aperture oi the horn is assumed to have a value b. The symbols illustrated in Fig. 4 have, therefore, the following meaning:

o represents the flare angle of the horn illustrated in Fig. 1;

b represents the horizontal dimension of the mouth or aperture oi this horn or the distance between the opposite horizontally disposed sides of the horn, at the mouth of the horn;

a represents the corresponding vertical dimension, at right angles to the horizontal dimension or the distance between the opposite vertically disposed sides of the horn, at the mouth of the horn;

Pl. represents the radial length of the horn, measured along one of the flaring sides, from the point or apex of the horn to the mouth or aperture; and

p represents the cut-ofl length, from the said apex to the free end of the throat of the horn.

These symbols will have approximately the same or corresponding meaning in the case of pyramidal horns two oi the opposite sides of which may be more or less parallel. It is convenient to measure the lengths a, b, po and pi in terms of the wave-length A; they may, therefore, ii the same unit of length be used for all dimensions and for the wave-length A, be represented by the symbols & 2 a A A A The space wave radiated by the horn in beams of different configurations, and the response of a receiving horn to waves arriving at diflerent space angles, depends on the shape of the horn in all cross sections, and the configuration of the exciting system at the throat, or 01' the wave delivered there by the hollow-pipe transmission line. It depends also on the flare angle o and the cut-ofl length Expressions for the field configurations and transmission properties 01' these waves may be obtained by solving Maxwell's equations in cylindrical co-ordinates and satisfying the boundary conditions on the surface of the born. The cylindrical co-ordinates p and o are shown in Fig. 12, in the XZ plane of Fig. 4.

In general, there are two distinct groups of waves: E-waves, having a radial component 01' electric intensity, but no radial component of magnetic intensity, in the horn: and H-waves, having a radial component of magnetic intensity, but no radial component of electric intensity, in the horn. Two subscripts are needed to define the waves of difierent orders. The subscripts n and m are used, each representing a positive integer denoting the number oi halfsinusoidal variations in the field between the top and bottom and the two side walls, respectively. Thus, we have the Hn n and the En,m types of waves.

For most applications 01 the horn, the H-waves are employed, particularly the two waves of lowest order, Him and Hm. The reason for this choice is that the configuration or the field of these waves inside the horn is such as to produce substantially single-lobe beams of linear polarization in the radiated waves. Both waves have constant phase on cylindrical surfaces about the axis within the horn.

Special cases may arise wherein several wave types may be used simultaneously. It will be assumed, however, that the throat or the exciting means ll, or both, have been so constructed. as described, for example, in my copending application, Serial No. 249,910, filed January 9, 1939, that either an Hm or H1,o wave alone shall exist in the horn, when used for transmitting; or so that the horn shall be responsive to the desired H0,1 or Hm wave alone, when used for receiving.

The Ho,1 wave may be excited by the currentcarrying antenna rod H in the throat disposed parallel to the Y axis of Fig. 4, as shown in Fig. 2, or by feeding an Him wave into the throat Irom the rectangular hollow pipe 18, as shown in Figs. 1, 3 and 55.

As appears from Figs. 5 to '1, the electric field intensity E in the Hu,1 wave, represented by the black dots in Fig. 5, is everywhere parallel to the Y axis, at right angles to the direction oi propagation. It is of uniform intensity in the direction of the Y axis, but it has a half-sinusoidal distribution in intensity along arcs concentric with the Y axis, such as the arc 1--I of Fig. 5, between the two flared sides, at right angles to the direction of propagation. The magnetic lines lie in planes perpendicular to the Y axis, or parallel to the X2 axis.

The H1.o type of wave may be excited by the current-carrying rod ll disposed centrally in the throat parallel to the X2 plane, as illustrated in Fig. 3, or by feeding an Hm wave into the throat from a rectangular hollow pipe.

In the Hm wave, as illustrated in Figs. 8 to 10, the electric lines of force lie along arcs concentric with the Y axis between the two flared sides; they have a uniform distribution along the arcs. but a half-sinusoidal distribution in the direction of the Y axis. The magnetic lines lie in planes passing through the Y axis.

These waves will not be more fully described herein because they will be understood without further description by reference to a paper by L. J. Chu and W. L. Barrow, entitled Electromagnetic waves in hollow metal tubes oi rectangular cross-section, Proceedings of the Institute of Radio Engineers, vol. 26, No. 12, December, 1938, commencing at page 1520, and also to a paper by Barrow, entitled, Electromagnetic-horn radiators, Union Radio Scientifique Internationale, No. 79, p. 277, containing a revision of a paper presented at the joint meeting of the said union and the Institute of Radio Engineers, at Washington. D. C., April 30, 1938. See also a paper by W. L. Barrow and L. J. Chu, entitled. "Theory of the electromagnetic horn, Proceedings of the Institute of Radio Engineersvol. 27, No. 1, January, 1939, commencing at page 51, and also a paper by W. L. Barrow and F. D. Lewis, entitled, "The sectoral electromagnetic horn," Proceedings of the Institute of Radio Engineers. vol. 27, No. 1, January, 1939, commencing at page 41.

Hamil" The dotted lines represent magnetic field intensities and the solid lines electric field intensities. The crowding of the magnetic lines near the throat of the horn indicates relatively large intensities. The wave-length in the horn decreases gradually as the wave travels outward toward the mouth or aperture of the horn. The waves undergo high attenuation and have a very small signal or group velocity in the throat of the horn. As they progress toward the mouth of the horn, their attenuation becomes less and the signal velocity increases. Near the horn mouth. the attenuation of the wave is that oi a spherical wave, and the velocity o1 the wave is that of light.

When the substantially transverse cylindrical horn waves reach the mouth of the horn, they become free from the guiding surfaces or the horn and spread out into free space as radiant energy along spherical surfaces. one of which, considered to be at a remote distance compared to wave-length x and the aperture of the horn, is indicated by a dashed line P in Fig. 12. Because there is no appreciable longitudinal field, these waves form a beam of transverse electromagnetic waves in space, and because 01 the limited dimensions of the mouth or aperture of the horn, this beam has a definite angular spread. The horn thus guides the electromagnetic energy from the throat outward in such a way that a substantially transverse wave is produced over the bounded, but relatively large, mouth or aperture of the horn. These radiation patterns, which are single-valued, show that the horn is unusually free of secondary lobes and stray radiation. and will operate well over a broad band width.

If the electric-field intensity E be plotted in polar co-ordinates, against the space angle 1, a radiation pattern will be produced in a plane substantially at right angles to the lines of electric intensity oi the waves within the horn, in which radiation pattern the value of the amplitude of the electric-field intensity E for any given 4 will be represented by the length p'. This will supply the radiation characteristic or pattern oi the horn.

The plane radiation pattern along the intersection of this sphere and the XY plane will be referred to as the vertical pattern, and the corresponding pattern along the intersection of this sphere and the X2 plane will be referred to as the horizontal pattern.

Let it be assumed that a receiving instrument is situated at the point P at a great distance from the horn, a distance so great, compared to the wave-length a and the transverse dimension a of the mouth of the horn, that the horn acts substantially like a point source. In polar coordinates, the point P is at a distance p from the Y axis, and makes an angle 4 with the X axis. Let it be further assumed that the point P is moved along the dashed circular are, at a constant distance p, but at different angles Q. At these different points P, of course, diflerent electric intensities E will be observed in the instrument, A curve is then plotted in polar co-ordinates, establishing the relation between the angle i and p the electric intensity E at the various points P. Such a curve is represented in Fig, 13. d =0 represents the horizontal line or the X axis, corresponding to the maximum electrical intensity E, when the point P is on the x axis, and any particular radius p represents the electric intensity E at the point P for the corresponding 10 angle Q. A simplified horizontal pattern for the 110.1 wave is illustrated in Fig. 13. Its forward axis 25 corresponds to the X axis of Fig. 12.

The shape of the radiation pattern depends on both the size or dimensions or the horn and its general shape. If the dimensions are altered. the radiation pattern will also be changed, and different sizes of horn produce beams of varying degrees of sharpness, Experiments have been made with horns of the type illustrated in Fig. l, the angle between the two pairs of opposite sides or which could be varied independently, and the length pl of which sides could also be varied, and the experiments were also supported by theory. As a result or these experiments and theory, it developed that, in evaluating the eflectiveness of a given horn to produce a directed beam of radiation, it is desirable to bear in mind three factors.

The first is the detailed shape of the radiation pattern, such as the presence and relative amplitudes of secondary lobes. The simplified pattern of Fig. 13 is shown without any secondary lobes.

The second is the angular width of the beam, or the "beam angle," indicated as such in Fig. 13. The beam angle may be defined as follows: Let an arc 2| be described, with radius 23 equal to an arbitrary one-tenth the length of the forward axis 25 of the radiation pattern, represent ing one-tenth of the field intensity. Let radial straight lines 21 and 29 be drawn between the origin 3| oi the radiation pattern and the points 83 and 35 of intersection between the radiating pattern and the arc 2|, The angle between the lines 21 and 2! is the beam angle. It is obviously twice the angle between the rorward axis 25. correspondingly to =0, and either of the lines 21 and 29, corresponding to a value of 5 which may be represented by The angle c. then, which is halt the beam angle, is associated with that value of the radius p that is equal to onetenth of the maximum value of E, corresponding to =0. The beam angle is a measure of the sharpness 01' the angle or the radiated waves. A small beam angle indicates that the horn is sharply directive. The smallest beam angle represents the optimum condition.

The third factor is the relative power gain. Assuming that a dipole were replaced for the horn, the relative power is defined as the ratio of the power that would be radiated from the dipole to that radiated from the horn to produce, in each case, the same magnitude or electric-field intensity at a fixed remote point on the X axis.

In the design of electromagnetic-horn radiators (and similar considerations apply tor receivers), therefore, two aspects of the horn are of fundamental importance. The first or these has to do with the excitation within the horn, as by an appropriate disposition of the exciting rod or rods It, or the desired type of wave to the exclusion of waves 01' other types. Reference may be made, once more. to my copending application, Serial No. 249,910, where the criterion is laid down that it is necessary to make the size of the throat and the radial lentgh 1 of the horn of such values that the desired wave only shall be produced for radiation at the mouth or aperture of the horn. The second important aspect of design concerns the radiation into space or a beam that meets the given requirements as to smoothness, sharpness and concentration or the radiant energy in one direction. To attain this end, it is necessary that two of the dimensions of the horn, in the plane in which these requirements are given, as, for

11 example, the flare angle rim, and the radial length pl, or the flare angle u and the horizontal-aperture dimension b, have definite values. By appropriate dimensioning, the beam may be made fan-shaped or cigar-shaped or of any other desired shape.

It was found that. frr a given horn length pi, there is a particular flare angle etc that yields the sharpest beam angle; it may be referred to as the optimum flare angle 4 for a horn of that particular length p1. Assuming a constant flare angle 950, on the other hand, one may flnd the best length 111 of the born for yielding the sharpest angle. It is possible, therefore, to produce beams the sharpness of which, in two mutually perpendicular planes, shall have any desired value. The exact relationships involved may, in any case, be determined from experiments made by varying the dimensions of the born. or they may be determined theoretically and lead to engineering design data on which the most economical-size horn may be determined and, in general. to definite information that allows the dimensions of horns for particular applications to be predetermined.

In Fig. 11, the relation is shown between the beam angle in the XY plane and the vertical height of the aperture a, for the two waves Hm and 30,1. No optimum is found in either of these curves.

The radiation patterns were computed by means of Huygens principle from the distribution of the Hertzian vector at the mouth of the horn, more or less as described in a paper by W. L. Barrow and F. M. Greene. entitled, Rectangular hollow-pipe radiators," Proceedings of the Instite of Radio Engineers. vol. 26. No. 12, December, 1938, commencing at p. 1498. The method, exactly as employed, was an application of a further development of the Huygen's method, described in a paper by J. A. Stratton and L. J. Chu. Physical Review, vol. 56, pages 99 to 107. July, 1939. This distribution was assumed to be the same as that which would exist at the mouth it the horn were infinitely long.

Experiments have justified this assumption for most practical cases. The radiation patterns were plotted both in rectangular and polar coordinates, and the beam angle was measured from the rectangular plots. The power radiated by the horn was obtained by integrating the Poynting vector over the mouth or the horn with the field at the mouth adjusted to give unity power density at a fixed distance and on the X axis from the origin, and was compared with the power radiated by a dipole to produce the same effect as before defined. Although the power gain obtained in this way does not include copper losses in either horn or dipole. it is believed sufliciently accurate for most purposes, as described in the said paper by W. L. Barrow and L. J. Chu, entitled, "Theory of the electromagnetic horn."

Over a hundred radiation patterns or composite curves of beam angle, power gain. etc.. were calculated for a wide range of horn parameters, plotted and analyzed. The horizontal radiation patterns shown in Figs. 14 to 29 show the general shapes and trends of all these curves. A number of similar curves have been produced experimentally. as shown in the said paper by w. L. Barrow and F. L. Lewis, entitled, The sectoral electromagnetic horn. and also in the said paper by W. L. Barrow. entitled. "Electromagnetic-horn radiators."

The vertical patterns, in the XY planes, as appears from the said Barrow and Greene paper, have the same shapes as those of a rectangular hollow-pipe radiator. This is because the distributions of the horn waves in the direction of the Y axis are similar to the distributions of the corresponding hollow-pipe waves in this direction. These patterns have a, principal lobe centered on the X axis and secondary lobes of relatively small amplitude. The sharpness of the principal lobe depends mainly on the vertical dimension of the horn mouth or aperture, or the dimension parallel to the lines of electric intensity of the waves propagated within the horn. The relation between them is illustrated by the curves of Fig. 11.

In these horizontal radiation patterns, in the X2 plane, for the Ha: wave. letting lEl represent the absolute value of the radius vector, from the origin 3|, of the polar co-ordinate plot, and the angle made by this radius vector with the forward axis 25 of the radiation pattern, it can be shown that the radiation patterns are given by the following relation:

B is a constant determining the absolute strength of the radiation.

e is tlibase of the Naperian logarithms,

1r is the ratio of the circumference to the diameter of a circle, and

J l(u) and .Mu) denote the Bessel functions of the first kind. of order Vz and respectively, and 0! argument u.

The vertical bars denote that the absolute magnitude is to be taken.

Equation 1 is a function of the variables i, 4m and a, which determine the dimensions of the horn, the wave-length 7\, the angle o made by the radius vector El in the said patterns, and the large distance p, at which the field is observed. Theoretically, the equation may be solved for o in terms of these other variables. Let that solution be expressed as follows:

In any particular case. p is fixed; A may be assumed given for some particular problem. with a particular horn having a length i and a vertical height a. The expression for 4: then becomes 13 The maximum value of IE], corresponding to =0, which may be represented by {mm}, is readily obtained by substitution. as follows:

will represent half the beam angle, as before defined. Corresponding to different flare angles. therefore, there will be different beam angles. and it may be shown that this function has a minimum for a particular value of 1110. determined by the equation a. Equation 1 will hold good for this value as well as for other values 01' when is substituted for q therein.

Similarly, the equation is m will determine the horn length p1 corresponding to the minimum beam angle. As will appear hereinafter, to satisfy this equation strictly,

but, as will also appear hereinafter, very little improvement is obtained beyond a certain horn length pi. As a practical matter, therefore, the shortest such length pl may be adopted in practice. For practical purposes, therefore, the lastwritten equation may be considered to be substantially satisfied when pi has any value equal to or in excess of the value before-mentioned.

Similarly, also, the equation represents the wave-length employed with any particular horn of length n and flare angle pa for yielding a minimum beam angle. Similarly,

corresponds to the minimum beam angle for a given ratio of horn length to wave length. Finally,

I iii-= will yield the minimum beam angle corresponding to a horizontal horn dimension or the dimension in a plane at right angles to the lines 14 oi electric intensity of the waves that are propagated within the horn. In view of the fact that. given the flare angle do, the horn is determined by either the radial length i or the horizontal dimension I). through the relation b i=9 Bill 2 the expressions d s W and m m are not independent.

It will be understood that it is not essential that the minimum beam angle be obtained analytically in this manner. It may be obtained graphically. For example, in Fig. 30, there are shown a number of curves showing the relation, for diflerent values oi.

between the beam angle and the flare angle do. In these curves, the expression before given,

dis

is represented by the minima of these curves. The minima of the curves in Fig. 31 would similarly be represented by As before stated, these curves do not, strictly, have any minimum, but it is apparent that these curves approach a limiting value as is increased and that there is no advantage in making greater than a fixed value.

The horizontal radiation patterns of Figs. 14 to 29 are not complete, even as half-patterns. because the mathematical analysis breaks down when exceeds a certain value. The experimental curves before mentioned, however, demonstrate that, in practice, the radiation patterns extend through the complete 360 degrees. Though the radiation patterns show slight irregularities, secondary lobes, however, are substantially absent from all of them. This fact, of considerable significance for some applications, is attributable to the half-sinusoidal distribution of electric intensity at the mouth of the horn.

Figs. 14 to 21 show how the beam-angle varies with variable radial length and a fixed flare angle n=30. When the curve is sharper than when yields a sharper curve still, and the same is true with the higher values of Beginning with about Pl 12 or 15 the larger values of BL A do not increase the sharpness of the curve. For practical purposes, therefore, for a flare angle u=30, one need not adopt a value of greater than 12 or 15, for substantially as sharp a beam will be obtained with this value of p i x as with any larger value.

Figs. 22 to 29 show how the beam-angle varies with fixed radial length and a variable flare angle o. The flare angle o=20 yields a sharper beam angle than a flare angle o=15. The beam angle corresponding to a flare angle o=25 is sharper still, and a flare angle o=30 yields a beam angle that is perhaps a little sharper than for o=25. From there on, however, increasing the flare angles gives a less sharp beam angles.

Sharpness oi the beam thus reaches a maximum with a flare angle on of 30 when When is held fixed, on the other hand, an increase in the flare angle o will yield, first, a decreasing beam angle and, later, an increasing beam angle.

In the particular value of the optimum condition was obtained around -o=30.

This can be checked by referring to Fig. 30, showing a series of curves, for different values of iving the relation between the beam angle and the flare angle e0. When an optimum value of flare angle o that will produce the sharpest or maximum beam angle. The beam angle decreases at first, and then increases; for very long horns, the beam angle approaches the flare angle 4m.

The optimum flare angle o and the corresponding beam angle both decrease with an increase ln horn length p1. For a flxed horn mouth or aperture, the beam angle decreases with decreasing flare angle and is a minimum when the flare angle is zero, that is, when the horn degenerates into a hollow pipe.

If the flare angle o is small, the beam angle of the radiation pattern is controlled essentially by the horn length pr or the horizontal dimension b of the horn mouth or aperture. Larger such apertures, for example, produce sharper beams.

If the flare angle o is large, on the other hand, the beam angle is controlled essentially by the direction of propagation in the horn, that is, by the angle of flare. The smaller the flare angle, the less divergent will be the directions of propagation in the horn, and the sharper, therefore, will be the radiated beam.

The horn length pi and the flare angle 4: have opposite effects. If the flare angle o is neither too large nor too small, the opposite effects of the two factors will compensate each for the other to produce a beam of sharpest angle.

Each curve in Fig. 30 has a minimum, with the minima lying along an approximately straight line (not shown) through the origin. which line deflnes the shortest horns that may be employed to produce a beam of a specified angle. The right-hand portions of these curves approach an asymptote at an angle of 45 to the horizontal. passing through the origin. The asymptote corresponds to a long horn having a beam angle equal to the flare angle on; the asymptote, if drawn in, would pass through the intersection of the lines representing 30 beam angle and 30 flare angle, at a 45 angle to the horizontal.

The same data is represented in Fig. 31, but with the flare-angle abscissa replaced by The curves, therefore, represent the relation between the beam angle and for difl'erent fixed values of flare angle etc. These curves have an envelope (not shown). The envelope corresponds to the shortest born that will provide a given beam angle. For example, a beam angle of 50 might be provided by a horn about 11 wave-lengths in length, which value is taken from the envelope. It is also possible to produce a beam angle of 50" by a longer horn, for example, by a horn of length equal to 20 wavelengths, having a flare angle n=l0. It is not possible, however, to produce this beam angle of 50 by any horn shorter than that given by the envelope.

Fig. 32 represents the same data as Fig. 31, except that the abscissa is here The curves indicate the relation, therefore, between the beam angle and the length will? 17 for diflerent values of flare angle o. If extended far enough, these curves would all approach asymptotes parallel to the beam-angle ordinate.

With the aid of Figs, 30, 31 and 32, giving the relations between the various parameters, it 'becomes possible to design a born that, for example, shall have the sharpest beam for particular conditions. As an illustration, let it be assumed that it is desired to design a horn as a radiator in a blind-landing system. Such a horn should be disposed as near to the ground as possible, not only because of economy, but also, and more important, because, if too high up, the horn may be struck by a landing plane. Not more than a given relatively small space can, therefore, be assigned on the landing field in which to install the horn. The height of that space would determine the b dimension of the horn.

Being given this height b, one could now, from Fig. 32, find the relation between beam angle and flare angle on. If, for example, the desired beam angle is 30", and b=)\, Fig. 32 gives the information that the necessary flare angle would be about 18. No shorter horn than the one so determined from the curves can rovide the desired beam angle.

It is thus possible, with the aid of these curves, to establish relations between these variables to yield the best conditions.

Fig. 33 is a composite curve showing the optimum flare angle on and the associated length IL A to produce a given beam angle. The curve of Fig. 33 is obtained by plotting the minimum points of the family of curves shown in Fig. 30. The optimum conditions represented in Fig. 33 permit the ready specification of optimum horn dimensions for a beam of given angle. If a beam angle of a given value is desired, the projection to the right of the curve shows the shortest length that can produce the desired beam angle, and the projection to the left of the figure gives the associated value of flare angle o. To illustrate, if the desired beam angle is 30, the optimum horn design would require a horn length 1 of about 30 wave-lengths, and a flare angle o of about 18.

Insofar as the left side of Fig, 33 is concerned, the curve is a straight line, just as is the line connecting the minimum points in Fig. 30. The linear relation feature does not exist, however, at the right-hand side of Fig. 33; a different nonuniform scale has there been adopted for Fig. 34 has a relation similar to Fig. 30, but plotted in terms of the power gain instead of beam angle.

The power gain for the Ho,1 wave is given by the expression and for the Km wave by the expression Fig. 34 represents a series of curves where the ordinates are P0,; and the abscissae are the flare angle die as determined by Equation 2 for difl'erent values of El A with represents maxima, instead of minima, and similar considerations apply to the derivatives of PM with respect to other variables. In some cases, it should be observed, these curves have more than one maximum, in successively decreasing order, so that the last-written equation should be understood as representing that maximum that has the smallest root. According to Fig. 34, an optimum flare angle, providing. this time, maximum power gain, exists likewise for a born or a given length yr. The maxima occur at smaller flare angles for increasing lengths of horn, and the magnitudes of the maximum power gain increase with increasing length p1.

Similar considerations apply to Fig. 35, which corresponds similarly to Fig. 32.

Fig. 35 shows, therefore, the variation of power gain with the horizontal dimension b of the horn aperture. As the power gain is directly proportional to the vertical dimension a of the horn mouth or aperture, ii the abscissae b were multipliedby for example, having an optimum flare angle of 14 and a vertical aperture the power gain is 720. A horn of these dimensions is entirely feasible and 0! moderate size at wave-lengths of, say, 10 centimeters.

The peaks of the curves in Figs. 34 and 35 provide optimum design conditions on a, power-gain basis.

The curves of Fig. 36 permit specifying, for both the Hm and Hm waves, optimum horn dimensions for a given power gain, and supplement the curve of Fig. 33. It gives information as to the optimum design for a horn to provide a given power gain; it specifies the shortest horn and the associated value of flare angle 4m that will provide a power gain. These values are specified in two curves, corresponding to both the Hm and Hm waves.

So much for the Hu,1 wave. Horizontal radiation patterns corresponding to Figs. 14 to 29 for the Hm wave are given in Figs. 37 to 52. Radiation pattern for the Hm wave in the X2 plane, i. e. the horizontal pattern, is given by the expression:

The same general trends are found in Figs. 37 to 52 as in Figs. 14 to 29 for the H wave, but the order of magnitude of the secondary peaks in the patterns is considerably greater with the Hi waves. For horns of equal length p1, increasing the flare angle 4, from a small value will, at first, sharpen the principal part of the beam and increase the magnitude of the secondary peaks. For suflicientLv large flare-angle values, these secondary peaks become larger than the principal beam, with the result that the beam becomes broader as the flare angle is increased. For horns of constant flare angle 450, the tendency is also observed for the beam to broaden as the length p1 of the horn is increased. For sumcientiy great lengths p1, however. the width of the beam is substantially equal in magnitude to the flare angle.

The explanation of the exaggerated secondary peaks in the pattern of the Ego wave lies in the uniform distribution of the field across the horn mouth in the horizontal direction and the abrupt discontinuity at the edges. The irregular shape of the horizontal patterns makes it impractical to define a beam angle for the Hm wave, so that reference must be had to the actual radiation pattern.

According to a. feature of the present invention, therefore, a horn may be provided that, whether or not under optimum conditions, a single-peaked radiation-pattern curve should be provided, substantially smooth. and substantially free of secondary lobes.

As appears from Fig. 4, for example, of the said paper by Barrow and Lewis, the radiation pattern is substantially free of secondary lobes for all flare angles below a certain value. It has been found that this value is approximately the same as that which yields the sharpest beam angle. If a horn is to be designed such that its radiation pattern shall be free from these undesired secondary lobes and shall have a smooth shape, its minimum beam angle should not be exceeded by more than say about twenty per cent of the value determined by the formulas given herein.

This value which should not be substantially exceeded, if a smooth lobe-free beam is to be obtained, may also be stated approximately in terms of power gain. It has been discovered that the smoothness of the beam shape of horns having values that exceed those from maximum power gain is also impaired.

As an illustration, referring to Fig. 30, a horn built for the propagation of the H04 wave at minimum beam angle may have a length equal to twelve and a flare angle o equal to thirty degrees. Because the curves of Fig. 30 do not have sharp minima, any value of flare angle to within about twenty per cent of this thirty-degree value, say between about twentyfour degrees and thirty-six degrees, will yield a very good beam angle. Horns of this same length to provide a smooth, lobe-free shape should have a flare angle c less than, say, a value that does not exceed by more than about twenty per cent this thirty-degree value. Any value of the flare angle o, therefore, up to about thirty-six degrees, though not necessarily yielding a. minimum beam angle, will provide a smooth, lobe-less beam shape.

To refer, similarly, to Fig. 34, an optimum power gain for a horn of length equal to twelve is also seen to be about thirty degrees. Here, too, a good power gain may be obtained by employing a flare angle o having a value to within about twenty per cent of this thirty-degree value. A smooth beam shape is, furthermore, assured for a horn of this length having a flare angle the value of which does not exceed, by more than about twenty percent this optimum thirty-six degree value. Any value of flare angle within about twenty-four to thirtysix degrees, therefore, will yield an optimum power gain and any value up to about thirtysix degrees will yield a smooth lobe-less radiation pattern.

For the H0,i wave in terms of the horizontal aperture the same principles apply with reference to Fig. 35. The same principles apply also, referring to Figs. 53 and 54, to the H1,o wave.

The power-gain curves of Figs, 53 and 54, for the H wave, are similar to those of Figs. 34 and 35 for the HM wave. The small oscillations of power gain at large flare angles o are associated with the shift of the energy from principal lobe to secondary lobes, as described above. The power gain is a linear function of the vertical dimension 

